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A 45 kg crate accelerates at 1.65 m/s^{2} when pulled by a rope with a force of 200 N. Find the angle the rope is pulled at. Friction is negligible.

68.2° above the horizon

Explanation

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Step | Formula / Calculation | Reasoning |
---|---|---|

1 | F_{\text{net}} = m \cdot a | Net force calculation using Newton’s second law |

2 | F_{\text{horizontal}} = F_{\text{net}} | Horizontal force component causing acceleration |

3 | F_{\text{vertical}} = \sqrt{F^2 – F_{\text{horizontal}}^2} | Vertical force component calculation |

4 | \tan(\theta) = \frac{F_{\text{vertical}}}{F_{\text{horizontal}}} | Relationship between force components and angle |

5 | \theta = \arctan\left(\frac{F_{\text{vertical}}}{F_{\text{horizontal}}}\right) | Calculating the angle in radians |

6 | \theta_{\text{degrees}} = \frac{\theta \cdot 180}{\pi} | Converting angle to degrees |

7 | 68.21° | Final Answer |

- Statistics

A 1509 g wood block is being pulled by the force meter at a constant velocity. Using the graph below find:

- The force applied to get the wood block moving
- The static coefficient of friction
- The kinetic coefficient of friction

Two blocks made of different materials, connected by a thin cord, slide down a plane ramp inclined at an angle \theta to the horizontal. If the coefficients of friction are µ_{A} = .2 and µ_{B} = .3 and if m_{A} = m_{B} = 5.0 kg determine

- The acceleration of the blocks.
- The tension in the cord, for an when the angle \theta = 32°.

*F* is used to hold a block of mass *m* on an incline as shown in the diagram above. The plane makes an angle of \theta with the horizontal and *F* is perpendicular to the plane. The coefficient of friction between the plane and the block is *µ*. What is the minimum force, *F*, necessary to keep the block at rest?

_{1} is stacked on top of block m2. Block m_{2} is connected by a light cord to block m_{3}, which is pulled along a frictionless surface with a force F as shown in the diagram above. Block m_{1} is accelerated at the same rate as block m_{2} because of the frictional forces between the two blocks. If all three blocks have the same mass m, what is the minimum coefficient of static friction between block m_{1} and block m_{2}?

In the diagram shown a 20 N force is applied to a block B (7 kg). Block A has a mass of 3 kg. Assume frictionless conditions.

- Calculate the acceleration of the system
- Calculate the normal force between the 2 masses
- Bonus: What would happen to the acceleration and normal force if the force was applied to block A instead

In the diagram below, A has a mass of 3.2 kg and B a mass of 2.4 kg. The pulley is frictionless and has no mass.

- Calculate the acceleration of the system
- Calculate the tension in the string
- If mass A is released from a height of 0.5 m above the ground, what will be its speed just before it hits the floor.

A space probe far from the Earth is traveling at 14.8 km/s. It has mass 1312 kg. The probe fires its rockets to give a constant thrust of 156 kN for 220 seconds. It accelerates in the same direction as its initial velocity. In this time it burns 150 kg of fuel. Calculate final speed of the space probe in km/s.

*Note: This is a bonus question. Skip if you haven’t yet taken calculus. *

^{2} N is being pulled up vertically by a rope from the bottom of a cave that is 35.2 m deep. The maximum tension that the rope can withstand without breaking is 592 N. What is the shortest time, starting from rest, in which the person can be brought out of the cave?

A train consists of 50 cars, each of which has a mass of 6.1 x 10^{3} kg. The train has an acceleration of 8.0 × 10^{-2} m/s^{2}?. Ignore friction and determine the tension in the coupling at the following places:

- between the 30th and 31st cars
- between the 49th and 50th cars

A student is watching their hockey puck slide up and down an incline. They give the puck a quick push along a frictionless table, and it slides up a 30° rough incline (µ_{k} = .4) of distance *d*, with an initial speed of 5 m/s, and then it slides back down.

Does it take longer to move up the distance *d* or back down the distance *d? *Or does it take the same amount of time?

**Bonus Challenge:** Repeat the problem but assume you are not given the initial speed, angle of incline, or µ_{k}.

^{2}. Find the friction force impeding its motion. What is the coefficient of kinetic friction?

An object has a mass of 10 kg. For each case below answer the questions and provide an example.

- If an object is moving, is it possible for the net force acting on it to be zero?
- If the acceleration of an object is zero, are no forces acting on it?
- Only one force acts on an object. Can the object have zero acceleration?
- Only one force acts on an object. Can it have zero velocity?

_{k} = 0.200. What magnitude of force should you apply to cause the box to slide down at a constant speed?

What is weight of a person who has a mass of 75 kg?

What is the mass of a dog that weighs 58 N on Earth?

An object weighs 300 N on Earth and 50 N on the moon. Does the object have less inertia on the moon?

Which of the following statements about the acceleration due to gravity is TRUE?

68.2° above the horizon

**A 45 kg crate accelerates at 1.65 m/s2 when pulled by a rope with a force of 200 N. Find the angle the rope is pulled at. Friction is negligible.**

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- Scroll Horiztontally

Kinematics | Forces |
---|---|

\Delta x = v_i \cdot t + \frac{1}{2} a \cdot t^2 | F = m \cdot a |

v = v_i + a \cdot t | F_g = \frac{G \cdot m_1 \cdot m_2}{r^2} |

a = \frac{\Delta v}{\Delta t} | f = \mu \cdot N |

R = \frac{v_i^2 \cdot \sin(2\theta)}{g} |

Circular Motion | Energy |
---|---|

F_c = \frac{m \cdot v^2}{r} | KE = \frac{1}{2} m \cdot v^2 |

a_c = \frac{v^2}{r} | PE = m \cdot g \cdot h |

KE_i + PE_i = KE_f + PE_f |

Momentum | Torque and Rotations |
---|---|

p = m \cdot v | \tau = r \cdot F \cdot \sin(\theta) |

J = \Delta p | I = \sum m \cdot r^2 |

p_i = p_f | L = I \cdot \omega |

Simple Harmonic Motion |
---|

F = -k \cdot x |

T = 2\pi \sqrt{\frac{l}{g}} |

T = 2\pi \sqrt{\frac{m}{k}} |

Constant | Description |
---|---|

g | Acceleration due to gravity, typically 9.8 , \text{m/s}^2 on Earth’s surface |

G | Universal Gravitational Constant, 6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2 |

\mu_k and \mu_s | Coefficients of kinetic (\mu_k) and static (\mu_s) friction, dimensionless. Static friction (\mu_s) is usually greater than kinetic friction (\mu_k) as it resists the start of motion. |

k | Spring constant, in \text{N/m} |

Variable | SI Unit |
---|---|

s (Displacement) | \text{meters (m)} |

v (Velocity) | \text{meters per second (m/s)} |

a (Acceleration) | \text{meters per second squared (m/s}^2\text{)} |

t (Time) | \text{seconds (s)} |

m (Mass) | \text{kilograms (kg)} |

Variable | Derived SI Unit |
---|---|

F (Force) | \text{newtons (N)} |

E, PE, KE (Energy, Potential Energy, Kinetic Energy) | \text{joules (J)} |

P (Power) | \text{watts (W)} |

p (Momentum) | \text{kilogram meters per second (kg·m/s)} |

\omega (Angular Velocity) | \text{radians per second (rad/s)} |

\tau (Torque) | \text{newton meters (N·m)} |

I (Moment of Inertia) | \text{kilogram meter squared (kg·m}^2\text{)} |

f (Frequency) | \text{hertz (Hz)} |

General Metric Conversion Chart

Conversion Example

Example of using unit analysis: Convert 5 kilometers to millimeters.

Start with the given measurement:

`\text{5 km}`

Use the conversion factors for kilometers to meters and meters to millimeters:

`\text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}}`

Perform the multiplication:

`\text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}} = 5 \times 10^3 \times 10^3 \, \text{mm}`

Simplify to get the final answer:

`\boxed{5 \times 10^6 \, \text{mm}}`

Prefix | Symbol | Power of Ten | Equivalent |
---|---|---|---|

Pico- | p | 10^{-12} | 0.000000000001 |

Nano- | n | 10^{-9} | 0.000000001 |

Micro- | µ | 10^{-6} | 0.000001 |

Milli- | m | 10^{-3} | 0.001 |

Centi- | c | 10^{-2} | 0.01 |

Deci- | d | 10^{-1} | 0.1 |

(Base unit) | – | 10^{0} | 1 |

Deca- or Deka- | da | 10^{1} | 10 |

Hecto- | h | 10^{2} | 100 |

Kilo- | k | 10^{3} | 1,000 |

Mega- | M | 10^{6} | 1,000,000 |

Giga- | G | 10^{9} | 1,000,000,000 |

Tera- | T | 10^{12} | 1,000,000,000,000 |

- Some answers may be slightly off by 1% depending on rounding, etc.
- Answers will use different values of gravity. Some answers use 9.81 m/s
^{2}, and other 10 m/s^{2 }for calculations. - Variables are sometimes written differently from class to class. For example, sometime initial velocity v_i is written as u ; sometimes \Delta x is written as s .
- Bookmark questions that you can’t solve so you can come back to them later.
- Always get help if you can’t figure out a problem. The sooner you can get it cleared up the better chances of you not getting it wrong on a test!